.

f,.. .. . . BECTIONC6PY. ‘ ‘ , -

-–—.-.,

. . ,- . . .., ..’ . . . .

. ..“

*

NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS. ..-

... .

TECHNICAL NOTE

NO. 1297

.

.

. ..“

BENDING STRESSES = To T6R=W

IN A TAPERED BOX BEANI

Langley Memorial Aeronautical Laboratory’:

““m:E?t3+ “::-’~;””’”:.‘.

.’.

. .Was~ngton

May 1947.

,.. .-, .-—“-.. .. = —-‘--“+>

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.,

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__ ...—.. —

~. ”.... .. .. . . .,.. .- llllgl~qg~g~~~lll[::::~~~..-:~,:. .,::, .;.,- .:..,.....:.J .. . . .. .. .:.

.- .1f.,

. . —--—-—~ -.— . . . , . . . - ‘ ..” ._== _- =.-—-— , ~. : :... . .. -——V= -----

- “~’&&’&~Rf CO*- ~R”j&&~&: ‘“:,-’‘.., ., .,. ,: :: “ -‘ ‘:...... .... .,-.. ... ., ,,. ... . -.. : .-

:b . . ...’ ;!...:?-. .: ....-. .. . ...:.... .“.,;. .-.:

..: .- ..,

,~~~~~~”N@~’~fJ; ~97 .: ““”“’ - :: “ ., , - ;.,. .

,.. ..’..,,-...., ,...,: ..... . :“. . .- ..--..”,..-“;’=ING”S Tkt@E& pJE m iORSIoN . ,“,””““ ““ : .“: “--‘.:>..

:,.’........:::::”’.. .’-:..“’ . . .,..” ~.. . .? --.,-. ..--,,:-...........?-.-, .‘,”. ....~:A”hW@~tii”.- .-. ‘“” .;, _ ._”z,. . .;,

. ..,”---.-’.“, .... ...

... i’.. “’;~y Ea&n””Tj ~&~gk~ , ‘ , “, :.:...;..,. .“ .:,,. -i

.. .~.”..:-;””..’...-1

.{.-—,... ....-... . ..... ..,-.. “-“-“SQMMQW ““ 1

,1

A meth&’.‘“is jpsented for thi “~lctiation ;f ‘b+iing ,itiesses . : . .due to torsion.ih”a ‘~~ered’bok’“beam..A 8p~cial ,*ger W* ,asmmed .. “:. ~,in which all Tilemgb?,.if extbnded”,would .z@ek“at.a.~int. we general::. .~ ,procedure:~f $rkly&ia gtvbn:,ik”similar ‘tp,#&, pr~cshre for a non- c

-1f

ta~ered beak pies6n~@. ~j Paul’I@hn in hi%:ja>eti.,~~,.~btkod..ofCalculating 3@13.ng sti+esies:D&e”t~ Torsi&j “*’NA$A:F$U?,,Dec. UJ42. .

..

Recurrene6.‘fmxml~ “.detblkqy%ifor use-in this”c~cul~tion are . “.. ./included.. A ccmqprison was made of flange”&d sheet stresses in” ,boxes with varying taperj.includi~ a,gon%=$~?red.~ox~ ,

“...:...:.:... . .-. ... ., -...— ______.. .. -......“.“-::; . ,,.-’

The resti%’ ckk@ie8 %y ‘%is kthod” iwire’&.pj@e5 wi~ exper~- . .. ~~mental data”dlkainsd f@m “teste”‘pkrfozmiqd..ona,ti~e*9d lox beam. 1The boa teatiwhs tested ukder.@“o i.@@W@n*. wiI-~++on8 of loading~first, d. cohckm%ti%d %orque at”th& tip M l.abti$,’:~concentrated ~

I

torque at the quarter point of the s~an. The experimen~l resultsobtained.from these tests showed good a~eement with the calculatedremil.te. Ca&cul.ationsfor the test sp~.cimepfor the two loadingconditiom. are also shown.

. ,. 1{ ~

,.

INTRODti~6ti ‘-”.- .::‘.-”‘:’””“’ ““‘“‘“ “ ‘-’‘---‘,........ :.+.... . ....:, ,... ... .1

The basic load-cam%ing struct~e .QfWW ..@i%Gra?t,~i%s is.a . , :box of approximately rectangular“:&&s’ eec%ion “ccin”sikfdiigof thefront and rear smrs and the tm ad bottom ekine of the *g c Thisbox is stressed by both bending-and torsional loads. Only tie

I

torsional loads are discussed in tie present paper. ;

The determination of the stress distribution in a box undert

torsion is a relatively simp3e problem provided that the crosssection of the box is no% constrained in any manner against warping.

I

I

.,. ..”. . .

,.. .

. . .

I

b

In this case the we13.-knownBredt formula for thin-walled torsiontubes is applicable. If some restraint is offered to warping,however, a sot of seoondary stresses is introduced in the box,Because the resultants of these secondary stresses are actuallybending moments in the planes of the walls and are accompanied bythe shear forces necesssry to cause spanwise variation of thebending moments, these stresses are usually referred to as bendingstresses due to torsion.

A method for the calculation of these bending stresses is‘presentedin reference 1. In order to simplify thecalcu~tion,Kuhn utilizes an assumption that the cross-sectional dimensionsand the torques are constant within each bay of the box and givesWe solution only for boxes in which the sides are parallel.Actual wings, however, ere usually tapered both in depth and inwidth. The present work is intended to furnish a theoreticalsolution of the effects of taper on bending stresses due to torsionand also to present experimental verification of the method ofcalculation. ..

The body of,this Raperis’ ‘dividedfnt&two parts,. The firstpart deals entirely with the4h60reticQ development of th8 formulasfor bendi~ stre&s6s,due.to:.@%M.on in a tapered,box,beam. Thegeneral ”procedure,ofsna~sispresented”ls “ai@lar. to the procedurespresented by Kuhn in,re?erence 1 fid by:Ebner in reference 2. Inorder to sim~lify,tie.~~emtical saalytii,s.a”.’&p8ciaJ.taperisassumed; “t@+’i@).the si,dgxs:’e$‘thab~x are ~“ssumedto taper linearlyin such a way as to meet at a point.

The second part of the paper deals with the experimentalverification of the theoretical formulkm. A description of the testspecimen and the test setup is given. Comparisons are then made oftho calculated and experimental results for two independent conditionsof loading* Complete n~r~cal soltitionsfo~ “bothloading conditionsare given in’the appendix. ..,,.

A

‘F

‘s

,. . . .,.

,.. . -“SYMBOLS ‘-

I ,. .“ . .

effective fltige areaj...squar~cheshes ““’ : . .,~.. .>.:”,. ..

area of flang6 angle, square inches..

area of cover stringer} square inches

i

i

1I

.1

●

.

1

r

lJACATN No ● 1.297

E Young’s modulus of elasticity.,psi .$ .

F flmge load at ~ point, pOUlldS

G shear modulus, psi

~1> %, KS taper constants

L length of flange in individual lay, inches

R tayer ratio(

% ;r .%bn.l Cn-1)

T external torque, inch-pounds

T volume of material, cu%ic inches I

x redundant flenge forcej pou@3 . .

s. distance %etween bulkh~ads, inches

b width of cover, inches

c depth of spar, inches

f, 8, P warping constants -

3

.,1

1

1

:

:

i1

II

I

I

I

designation

designation,.

shear flowj

designation

‘ distance in

I

for general temn in series I

for typical lay

pounds per inch

of root bayi

each bay measured along axis in tlSD@j il?ohesI

i

sheet thickness, inches

warping deformation inches

distance in each bay from outboard bulldaeadmeasured i

along axis, inchesI

angle between flange and center line of cover, radians

1

1

1

4

UC angle betveen

o normal flange

T shear stress,

Tav

average value

Subscripts:

‘iflange and center line of spex, ~ai~8 I

stress,‘psi ~ i1I

of sheer stress, psi

,.

b refers to covers

o refers to spars

i designate inboard

.

end of bay

o designates outboard end of bay

Superscripts:

I,

IiIiI

\iiI

‘11

T designates stresses due to torquer, I

,I1u designates stresses due to dummy unit loads

.;

x designates stresses due to X-forces iI

The subscripts of the redundantdimensions b end c designate thewhereas the subscripts for T, w, p,under consideration

fbnge force X ma of i

stations at which they exist,3, and f designate the bay ~

The bulkheads or staticms are denotedhy O, 1, 2,... n-l, n, .

Ml,... r, starting from the tip or out%oard end and proceeding tothe root or inboard end. (See fig. 1.) The bays are also nuzibered . .from the tip, the tip bay being designated as number one (Beefig. 1)* A bay therefore carries the number of its inboard bulkheador station.

8

. . . .

.t’

. .. .., .,.. . .

1 “ “

I

r

H4CA TN No. X297

IfEVELORYIPIIWOF TBEO~TICAL FORMULAS

“ El actual wing design neither the sp.nri’sevaxiatibn of’‘&etorque nor the cross-sectional dimensions cszibe represented.%y-simple mathematical expressions. In order to simplify the

...,

5II

I

Imat~bmatical calculations the box-is divided at we bizlkheadsintoa-number of hays and We torque is assumed cons-t within eachinQ+yidual bay.

I

.. -.,.A box beam under %ortiianis an indeterminate struc%ure. B I

odder t~ mike the structure statically determinate, the box is cutat each bulkhead end redundant flange forces X are applied atall flaiges.. These X-forces az% axial forces applied at each

iflange, as shown in figure 2. Ihder the acticn of the torque Tand the flange forces X the %OX is deformed as shmnl by thodashed lines in figure 2. The amount of detonation is calculatdby the use of the princiyle of vfrttil work, sometimes known asthe duumy-unit-loed method. The X-forces are then.found by the

I

< application of the principle of consistent deformation of edJacent tbays.

4

b

* Sim “.convention.- External tor&es T are positive when actingclockwise as viewed fron the tip. Shear stresses T arb.~sitivewhen acting in.the di.rec%ianof shear s.tresse.scaused by positivetoyque. The X:@ices.ar@ po~ittvq when acti+@ irithe directionsshown in the~ketch” iti”figg.ye“2. ~ormalptresses “’a are positivewhen caused By”positive X-forces. The w5mpZi3gdefbrmat~oti”ti--is

.7-.-—. ......

positive @ we direction aq shcwnby the:dashed lines in figure 2.., . ...—.. ..,. .. . . .. .”...:.-.-., , ..

@neral &umntt’6ns.y:The cross section d~ the %6x is &&tied to....‘berectsn@.er.knd doubly.syrmmtrical;:“The shap5 of the cro&s sectionis maintained by “t+.eMlkheads, which are l%simed”ko be rigid in.their own planes. h place of theaetual wtructie,’tbe equivalentstruct~e.s~owr+ in figure 2 is used, @ which all the area capableof Kzy!yZng norpal~ptress’esis concentrated i~’the”flanges.”“-”TheWEWS ~f the equivalent’s~uciv~e are.ass@e&%q car@. only shearstiesses .@adthe”flanges;.all the.norinal’strqss”es.“,In’crderim

.. -. allow for the fact mat the waQs ‘can”acttilly’carry&5M&%2 stresses,e.ach,fl~ge,”area is increased by ~one-sixthof”the emea of..bothacover and a spsr web. ‘Ifth& covbrS incl~e &r@ersj-d effective5tringer area is aadea to each flange. This effective stringer e+wais that are~ which, when concentrated at the flemge, gives the samesection modulus a%out the neuia?alaxis of the cover as the actualstringers. In the case of equally spacsd stringers the effective

}.!

1

. .

I1

I

I

str~er area istaper of the boxeffective flange

simply one-sixth of the total str@er area. Theis such that the flanges meet at a point. The

II

area is assumed constant witiin each bay.

NACA TN No. X297

Stresses in “hnIndividual Bay

The formulas given herein are”derivc%ifor a ~ical bay n,boundedby thelmlkheads n-1 and n, (fig. 2), The bay is acted I

on by three independent eets of loads:, a torque Tn on both ends

of the bay, a group of Xn-forces on the,inloard end, and a group

of Xn.l-forces on the outboard end. Formulas are derived for the

stresses due to each of these independent loads. The final stressdistribution may le obtained by euyerposing the individual stresses.

Stresses caused by torque,- The shear stresses cs,used by thetorque acting on a bay are given by tho well-knoti fozz?mlaforshells in torsion (Bredt’s formula)

Tn 1.

‘b = 2bc~ ..‘> ““ .I

. (1] , ,

_ T’n”T

c 2bctc ‘ Jr,

where b is the width of the cover and c the depth of the sparat some distsnce x’from the,outboard end.(fig. 3). In the case ofpure torque no normal stresses are set up in the flange.

S&eases caused by the X-forces.- When a set of X-forces isapplied to the end of the bay, both axial stresses in the flangesand shear stresses in the walls are set up within the bay. Unlikethe shear stresses for the nontapered box, the shear stresses in thetapered box are not constant throughout the bay. : .

In order to study the distribution of the shear stresseswithin the individual walls, asection of a wall is isolated asshown in figure 3(a). The free body shown in this figure is a part

of a spar web bounded by two planes cutting the spar Just insidethe flanges and by,two parallel planes, one just inside the n-1 bulk-head and enoth6r parallel to it and at a distance x from it. The

,,. .

h

I

.

1

I

I

m

,>

. !:

I

c

... ,,. !,.

““loadingon the’body is also shown in the figme.—

By sunmation of - bmoment iabout the ,pointof “intersectionof the flanges, en expressionis ob%ined for the ~hear flow qc in the spar in terms of the out-’hoard sheer flow q%-l

1

(2a)

II

..

ad, similarly, in the cover ,

2.

H

bn-l, qb..!l~=-b n-1

(2b)

-.,

.’ . .

.“

4 :. .i ,.”

where tlienotation is

., .rt

the same as -t show

of the shear flow ~

of a spar web isola~~

in tigure s(a).-1

is obtained from an IThe distribution

infinitesimal section as shcnm in fQure a(b).The free body shown in this figure is,a sectiionot the spar webbounded by two parallel planes an infinitesimal diwtanee @ apartand by two planes cutti~ the spar just inside the flange. From

.,I

a summation of moments, the fundamental shear-flow,relation in thespar for a tapered box beam is obtafne’d,

1,qc = qc’

x(Sa)

I

1

and, similarly, in the cover. .

‘.

,

;(3b)

.1

shows that at every point along the box tho shear flowflemge %s equal to the shear flow in tie walls at

This equationacting on thethat ~oint.

.. .’

\ s’- ‘ In orderthe cover end

to ~btain a relationship lwbween the shear forces inin the spars, a free body of a cross section of the

6,

L -.

.

,I

1

.,

box iu considered. Since the taper of the box is such that thecenter lines of the fl~es meet at a point, the flange loadscontribute no torque and,the condition ZT = O gives the equation,

qb~c + Qc = O

or

q~ = -(it (4)

Two expreseicms for the flan~e loads are derives, one for theXn-forces and another for the Xn-l-forces. The free body of the

flange in figure 3(c) shows the loading of the flange Unaer Xn-forcos

only. A summation of forces along the flahge shown in figure 3(c)gives an expression for the flange load F at any point along thebay:

11X fix

Ias ‘ dsFti“— q I“c ‘+-l! /’ ‘bx

ax=o

LJO x do(5)

where ~~ is a constanti ax

combination of equations

r:t1

.,

,1

m

●

1

L

tiepend.ingon the taper of the box. The.

(2] to (~) gives

,,.&

,

I If the integration is perfomned with c given as a function1of x by the equation

I1

C=w[’.-%ad%ad~ b

n

,i

c

NACA TN No ● 1297 9

L

equation (6) becomes

F= ‘%-l x2L ~cn-l --- a (8)

I

I

.

where, L is.the length of the flenge and a is tie distamce betweenImM&eads. , I -,

t

With’ the forces Xn-l applied at the outbar”d end of the bay,1

fMm.ge loads become1.

the

(9) I

In order to o%tain ~ In terraEIof the X-forcos, the valuen-1

F for both Xn-l-.!

and Xn-forces is calculated for x = a..,

of

For the case of the ~-forcest.

,. .,Y

x= ‘%-1 ~ “=2(J —‘n-l ‘%

I

I

or 1

-,

Xn!lC = ‘~b =R—n-l’ n -1 2L ‘

,

—.IFor the case.of the Xn l-forces--

,“

0 Cn-l ~=,xn-l + 2q -—Cn-1 %

-t--

or

x~c = ‘qb = -R ~n-1 n-1 2L

.

(11).

I

.,

10 NINA TN No. 1297 .

bnwhere R is the taper ratio =

%-—. = —0bn.l

.c~.1

By substitutin~ equation (lG) in equations (2), (4), and (8] andequation (11) in equations (2), (4), and (9), a summary of the stressesfn the bay in terms of the X-forces at the end can be made. Forthe Xn-forces

x 7cncn~ ~

q= ‘“—-X ~2 2L I

!& .?!3%x=-# ,.}

F Cn X Xna =—=.——.—

A caA J

and for the Xn-l-forces

Cnon-l %-1q= - — .——Cx c2 2L 1

I)nbn,-l Xn.lqb =— —x b2 2L

where A is the effective area of the flange.

(12)

(13)

andits

Defomnation of an Individual Bay “ ...

Frtnc3731eof calculation.- Under the action of the torque— ——grouys of X-forces the cross section of the box warps out ofplane, as shown in figure 2. The magnitude of this warping is

calculated by the method of virtual work: The following three-stepsare necessary to obtain the _tude of the warping. First, thestresses T and u due to the a plied loads are obtained and

second, the stresses t?# and a due to a system of dumgy loadsare calculated. These dummy loads are unit htaf3 applied at thepoint where the deflections are desired and.also in &e directiondesired. The last step is to obtain the defamation by tie of theequation

,where V is the volume of the stressed material.-%ased on the yrticiple that the external work donemust equal the internal energy stored by virtue-oftho unit force.

(14)

This equation isby the unit forcethe existence of

Examination of figure 2 shows that the warp- is doullyantisymmetrical end consequently the dummy loads employed in tiesolution ~e also doubly antisynmetrical. This group of tit loadsis stilar to the group of X-forces and therefore the formulas forthe X-forces can be,usedin the calculation of the stmesses causedby dunmy loads.

~arping caused by torque.- The stresses caused by the torqueacting on the box are, from formula (l),

L3=o

Tn.”~ “.

b= 23c~

TnTc = —-

2bctc

1/J

(15)

I

\“I

i

I

1

1t

,,,

I

I

1

i

i

!

32 NACA TN No* 1297

.,In order to obtain the warping in the nth bay at bulkbead n, theantiqmm.strlm.l group of unit loads is applied at the inl?oariie?+i.of the bay. The stresses caused Ey -theseforces axe calculatedfrom formulas (12) by placing Xn = -1. The results are

,, ,,

\

. .

,(16)

? ,,;U.. = “ %&A_L_c“ ~2 2Ltc.,

J ““...,,

.

.for v, and Uu TJTbJ ‘c and for a , Tb Y md TcThe results

given in equations (15) ati-(16) are now substituted,in equation ilk)to give

1

. . ,.. .$, ~wT=,2:. -:-(”%=JC’A-.‘ “f}&G”l ~..

n~/o .G2bctc

,., ..

L“’%%?+%)!!!”? ~

,., ..,. .J. ‘+ 2:.,,. .,. . .., ,,. ... ,. .. ,,

,.

,.. .

.’

4

,

I

ii

t. . . ,. ,. ,... . .

which yields when integrated..,. ,,,.,.. .,,.

.

,,, ...

ab bn.J 1+-R— ——.. “%cti — ‘(17)

L .% L ~c ~

T Tn‘ni = ~—-

n-lcn-l

I

.

,,

1.

,

I

,I

.

,NACA TN NO.3297 13i

.I

3iIorder to “obtainthe warping due to torsion at bulkhead n-l,the groups of unit loads are applied at the outboard end of the bay.Now the stresses due to the dummy loads,.aracalcxbtsd fromformula (13) with ~.l = 1. The results are

1

as fOllowe:

u“T. =.- %2Q 2.c ~2 2Ltc

(18)

I

.,.,.. .

I

1

.The stresses due to the torque acting m

show in equation (15). An inspection.of tieand (18)) caused by the unit loads gives

i

the %ay are thosestiesses (equations

Tw’ =W”o %* = ‘“T

1

(19) I

I

Warping causea,b~ Xn:force$.A me wexping

of tie “bay”caused by the ~-forces ZS o%tained

at the inloard end

by the a~plication

of equation (14) in the foti .I

x~4wni =

% TbT#4 /%!! A ag”+ 2s

.G b~qJoE,,~

J

‘u ““ “.+2 ‘aCTcTc

— Ctc .(IXC-o~.,

.. .

,[20)

ISubstitution of the values for the stresses due to the unit loads(equation (16)) ma the stresses due to the ~-forces (equation (12))

. in equation (20), causes.the equation for wa2ping to becoms.

I

%1. ‘ni = -pnxn (21) t

1

I

t1

t;

iNIKX TI?”hobX297lk

where

*

ti

i

iii

1

,,. . . . ... . .:, . . . .. $,: .,,. . .“

. . . .,, : . . .,.> -. . . .,.. “: :.! ): .,” :-. ”

.’, .,, =,. . .,,.

(.....,-.,.,...:,..,,,.. .ab’bn:l~~~~?l,. ‘~~ ; R.....“ ;,

la-l,,’,~1, g’+.+L,= ~‘Ltc “) (22)

,.,,.

8. .

,.. :,...,”. ,

and#2 (;2.1

)(R -.1)3 ~ ‘2 loge R,Kl= (23 )

,

The

the

CO.UStSrltK1 iSindividual bays.

de~endent entirely on the taper ratio for...’ ‘. .“. i

1

the taper’“ratioapproaches unity, theexpression for Kl Q equation (23) becomes too sensitive for

1

:

.,

As the value of

practical.use. By expanding.t@e logarit~.lnt~..m.i.nfini% seriesof’ (R -“2), ,“.’ ,.

.. ’., ,,” .,

‘1 =1+$(R ‘1) ‘;, (R ‘1)2+R -1)3 ,...

. .... .,’. . .,

WI g+’(-1] (R “ l)m {24)., (u +..,l)(m.~,2)(U i-~,

., J......... ..... ... .. ............,... ., ,.,., .. .’

.n

I

This expression can only he used for:..smallvalues Of ~R, bince theC3erlesconverges only for values of R < !2, The rate of convergenceis very slow, however, after R reaches a value of approximately 1.5.For values of R <1.7 orily:fqur.tbztiof ’theseries are needed toevaluate K1 within 3.percent. A,graph of numerical values for Kl

from R = 1.00 to R = 3.00 can be seen in,fi~e 4.,.. . ..

The warping at the outboard”end of.the bay caused by the~-forces can be written in the form

,,, ., .-. xi. j,, ::. .: ., . . .

. w~ =-J&,:”’,’, ‘ , . . . .. . .,, . 0 :,, (25). . ... -.!... .

. . ,

J

,’..”

1

I

IQWA m No. X297 15i

I

,

,

where the coefficient J=, is obtained.by the application of

equation (14). By substituting the values for unit stresses fromequation (18) and the v~ue for xn-s~osses from equatim (12) intoequation (14) and Integrating the result, Jn is found to be givenby

(26)

Iwhere . .

[

2R2 -2R-(1+R)R1O$3*R

%? =-6(R - 1)3 .1

1

(27),

I

I

For small values of R, again the expanded ‘seriesfor K29

I●

1

m+l 6(IE-L)+ (-1) (R - # (28)

(m+l)(m+2)(m+~)

is more ~ractical. As for Kl, the rate of-convergence of the

series makes the expression pract~c~ only for the range from R = 1.00toR = 1.50. The numerical values for ~ are plotted in figure 4=

.. .

.1 I

Warping caused “by Xn-l-forces.- The warpin~ at the inboard

end of the bay, caused hy”the Xn-l-forces, can he shown, by meansI

of l@xwellls law of reciprocal deflecti&j to %e equal to,. ’.. . . . .

,., . /%1 “’”’”““ “’””” .,=Jnxn-l :(29)

% , , -i

. . . [

where Jn is the expression given in e~-tion (26).I

iI

.’

1

.

i

3.6 mm m NO.1297

The war~ing at the out%oard end of the bay Is derived bysubstituting the values of the stresses for unit loads fromequation (j.8).andthe,vtiluesof the stresses for the Xn-l-forces

from equation (13) into equatfon (14). Upon integrating the result,the following expression is obtained:

I.,

Xnq “%0 = fnxn.l ‘

. .

where.,

and

r(R2 ‘i)-~lO&~l

1.. , -w.. . —

‘3=3I

(R -

For small values of R, the expahded

‘3 “=-#-(R-I)+~(R-

10

(30)

series form for K,3’

~ (R-1)3...1)2 ---

+@)m— 6 .(R -l)m.h ..

(m+2)(m+~)(33)

should be tis”ed.‘As was the case for the exp&ssions for K1 and ~,

the rate of convergence of the series makes the expression for K3

: practical only within the limits of R = 1.00 md R = 1.!30.

Comparison of tanered-box formulas with uniform Cross-seotiOQ$Ormulas,-A comp~ison of the formulae derived forthe taperedbox beams with those formulas derived for the nonti~ered boxby Kuhn in reference 1 can easily be made by obtainin~ the formulasfor the special case of %ap~r where the taper ratio is unity.

1

1

.,

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, .P

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m

MICA TN No ● 1297

,

17.. ,.

following relations hold trueWith this assumption, the

a+ % C=L=a I

I.,.,-,

,..,

Equation (17) now becomes

1

I

.( .)TT%

&bctb-f .‘n= —-—,I

,. ... .,. _...,...

which is identical with equa%ion (21) ~f referenge 1. Equations,(22)and (31) become -. . .

I

i

which is identical with equation (23).of reference 1. Equati~ (26)4 ~

becomes [

(J~ = -& +&*.”:. +.c )~.’””’ . ..:

1

.!

.. . . I

reference 1:

. .

which is identical tiw equatio~ (?7).o.f

This comparison shows that &ll.the t&ered-box formulas for thes~ecial case of R = 1.0 or uniform cross seCtiOn are identicalwith those formulas developed in reference l.’ “

privation of a recurrence formula.- The total warpi~ at theends of the bay n due to the combined effects of the three 8eP-teforces T, ~, and Xn-l can now be obtained by a sption of all

the otiponent parts. The equation for the warping at the inboard end of‘bay n is given by the sm” of equati~s (19), (21), ~d (29).

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1

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i(34) ‘

t

The equattcn for warping at the outboard end of bay n given ~y./’

the summation of equations, {27), and (30) is

T‘n. = Wn - SnXn + fnXn..l (35)

The oquatzionfor warping at the out%oard end of bay n + 1 isobtained.merely by increasing all subscripts of equation (35) by . “one and is

(36)

Aocording to the principle of conoi.stentdeformation, thewarping at the outboard end of one bay must be equal to.the &wpiq&at the inboard end of the adjacent bay. A recurrence formula cantherefore be obtained by equatir~ the warping formuJ..asof twoad~acent bays. By

and the expression

recwrenoe formula

equatin~ the ‘~xpressio~for Wn: in equation (34)

‘or ‘(n+l)o

Iecomes

.!.

in equation (36), the general

‘.

‘ +&l - ( “T-VT(37)‘n+l + %,) % + ‘Il+lxn+l= %+1 n

By giving n successive values from’ n = 1 to n =r, ase% of- ; equations is obtained, each of which contains wee of ,..the redundant X-forces. These equations represent the continuityconditions at stations 1 to r. The tip of the box is usually freefrom any restraint, thereby making the farce X. at the tip”of the

box zero. Therefore, for a box divided into r bays, r redundant .’forces and r equations exist. #

Boundary cacklit~m..-With the tip of the box free from my ‘restraints end the tip force X. = 0, the first equatlcm of the . .

system is “.,.. .

,

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i11

,11

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F(38)

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NP.CATN No, 3297 19iI

When a .heam is at@ched to a rigid fo~~tion, the foundation maYbe considered a bay r + 1 ‘having izifinitesheti stiffness and

I1

infinite e=iialstiffnes.s;therefove, ‘“,,,

The last equation of

TT”=fr+l r+l

.jwi.~ 1

ii

the system now becomes ,

(39)

~omparison of stressesfor “boxeswith ya&yinK taper ratio and~lawe area.- In ~d~to show”the effect of taper on the bendingstresses due to torsion, calculations were “madeof the flange qnd“sheet stresses for b’oxeswith varying taper ratio and flange area.‘-AU boxes were.120 inches long end were divided into six bays ofequal length. The yoot sections of the boxes were identical. Thedimensions for the boxes.at the root and ti~ can be seen in t&ble 1.

In figure 5(a), curves of the fLange stiesses me dxa~m forthe three boxes with constant flange areas and also for themoderately end highly tagered boxes with a tapered flange area.“In-figure 5(b), curves of the sheet stresses are drawn only for thethree hexes with the tapered flemge area. .,.

Figure 5(a) shows that the fhnge stresses for the case of themoderately tayered box (curves C and D) are only slightly greaterthan those of the nontayered box (curve E) . Examination of the rootflange stresses for all.cases shows-that the increase in thesestresses for en increase-in taper is very s~l~; me root fl~gestress for the highly tapered box is approximately 10 percent abovethat of the’nonta.yeredbox. Since, ae in the nontapered box, theflenge stresses for the box with moderate taper decrease ?ery rapW1.y,the only appreciable.flange stresses occur in the vicinity of the root.As the taper becomes greater, however, the flange stresses alcngfietotal spe.qincrease. Insyectlon of curveq A and B shows that -the flsmge str’essesat a p@nt, 20 inches from the tip =e approxi-mately one-third of the mextium flenge stresses “at the root.Consequently, for the box beam with moderate taper the bending stressesdue to torsion i.mthe outboard end are negligible as compared withthe stresses ne’arthe root, whereas in the highly tapered box theflange stresses do.}ecome,e.ppr@ciab18. A similar cmclusion for thesheet st@sses caq%e T,eached,%y~e- $@pection of figure q(b).

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20 NACA TN 1~0 ) ~97

Comparison of curves A and B and curVes C and D infigure ~(a) shows the effect of a variation of flange area alongthe span on the flange stresees. The maximum effect apyears nearthe tip of the box where the fla~e streus for the box with tapered-flamge areas is approximately 20 percent higher than the flangestress in the box with constant flange area.

Two sets of calcn.alat~onsfor the root stresses were made fora moderately tapered box under a distributed torque loading. Thed.istiibutionof the torque was such that the increment of torquein each bay was pro~rtional to the chord of the hay at the inboarden~. The method presented herein is Used’for the first calculationwhich was an exact solution of the root stresses. The secondcalculation was made on the assumption that the box consisted only ofthe root lay with the total torque acting on tie outboard end. Theresult of these calculations showed that the rcot stresses calculated‘bymeans of the exact method were approximately LO percent ~eaterthan the root stresses calculated %y the approximate method. Similarresults were o%served for a calculation of the stresses in the highlytagered box under a similer distri%ut~,onof torque. TWmn only theapproximate value of the stresses at the root is required, a satis-factory answer can be obtained by assuming that the box consi.wtsonlyof tie root bay end that the total torque is concentrated at theoutboard end of that bay.

In both sets of calculations with the distributed torque, theflange stresses near the tip of all the boxes were approximately ofthe same order of magnitude as those of the moderately tapered boxwith tip loading shown in figure 5(a].

EXPERIMENTAL 9EKIFICATIC)N

Test specimens.- In order to obtain.—of the formulas derived herein, a large

OF FORMULAS

experimental verificationtapered box beam was

constructed and tested. The bo~ was &de sfimetrical about the midspanand was supyorted there by a rigid frame as shown in figure 6.Because of this setuy, complete re~traint against warping at theroot, which is the midspan of the box, was assured. Equal torqueswere applied at the tipe end later at the quarter points of the span.

The material used for the %OX wss 24S-T aluminum alloy and thegeneval dimensions of the specimen are shown in figure 7. Thethicknesses of the covers and the spw webs and,the sizes of theflanges, stiffeners, and stringers me also shown in figure 7.General dimensions and.atcringerspacin~ of both the root and.

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NACA TN No. 1297 21

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tip sections are bhotm ir”fi~e 8, along with the &mensions for,.

the equivalent sutmtitute”se”ctfons,.-,-

The dtien?ione of the boxvary linearly from the tip to the’root,’-‘-whereasthe thicknessesof the sheets and sizes,of the flanges and stringers are constantfor the whole box. “Although the flange ”arba’otme qimp~ifieds%ructtie’Vsrie& .line&l.yfrom tip “tipto.& robt, due”to the .addition of one‘~ixth the area of the”cover anil8P” we~, anaverage flemge area in each ~ay is used in the calculations,.,Inorder to asstie thqt ~e”%’timea,ds“were-fi$.gidfdr q~.?~~ctical

purposes, the bull&eads were,made of-fozzned%.mhsteel plate. ..8. . . . -.,

Test “set~~.- The.general test set&’ is”showr-”in f@ure 6:~ :..”The &x beam was comecl=d to tbe.cen~r ~wer”.W m=~ of fo~ “.steeJ.flexure plates, one”on-each side of,the box. In order~to-reduce the end effects as much as possible, the box was connected

to the flexure plates at the root by closely spaced ~-inch bolts.. .. ,,:’

This t~e’o.f connection permits the torque-reaction to.be distributedas a uniform shear flow.arowd the ~erimeter of the box. Figure 6..shows the loading arrangment.at the.$ips of the box. The ssme ‘method of loading was used when Me tests-were run.with loads at .~e..quarter-s~,,yotits. . :, ..’‘. ... ‘ . .. ‘:..-’

. ,. . . . . . . .... . .. ..; Test wmcedure.- Strain surveys were qle.defor bo.tl-’loadingswith 2-inch. Tuckerman optical s.tm.ingages.” Shear.-strain measurem13ntswere taken around the perimeter of the box at Sectl’onsnear the -, -.

c.q~te:. l-h? Of each.bay, @ also at,~ections 14,2 ?.c~~ Q? e~:yerMe “~she&’.%t&& measmements ~cros~ .side .ofb&ead8 ~’A “~:

any cross section consisted in measurements made .%etweenthe . .stringbrs and between the.”flange“’and adjacent stringer on the coversend three equally spaced measurements in the spar wel. The shearstresses were obtained from strain readings at 45° and 135° from theaxis of the structureL””The no-l. strafis were me~s~ed alon$ each

the rOOti “’ .. .. . . .“ :-:

For each,teet run, strain-gage readings were taken at zerotorque and.a~ter each of four equal increments of,75,000 .tich-poundsof torque. The load was the~ released end another zero reading w%taken. If the two sets of zero readings did not agree within 100 psi,a new test run was matie. The strain readings for each gage were thenplotted against torque end the best straight line was drawn.throughthe points. If the line did nat”intersect-ae Qrigin, a P@mllel Ltiewas drawn.throvgh the Qri$tis If, however, the new L1.newas displaced

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22 NACA ~ No. ~97

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from the original-llneby more’than a strain equivalent to 200 ,psi,a new set of readings was taken. The values of stiain were thenobtained from this new parallel line.

Test results.- In converting the results obtained from the ;:,sta%in surveys to stiessesi Young’s modulus was taken as 10,600 ks’iand the shear modulus as 4,000 ksi.

.’

The observed sheti stress of a cover or a spar at any section ‘Is the average of the shear.stresses obtained for the two opposite “oovers or spars at that section. Figure 9 shows that the distributionof the shear stressesacross the covers and spars is oonstantthroughout the ‘section,except for the part of the skin betweenthe flange and the adjacent stringer. These plots substantiate in -yart the original assumption of uniform shear stresses over thecross sectionc

The observed flange stress at any section i~ the average oftie flenge stresses obtained for the four flanges at that section.In figure 10 a plot of the stringer stressesat a oross secticmin bays 5 and 6 is shorn. Strain readings were taken on eachstringer on the leg adjacent tQ the coverc The cross sectim in ~bays 5 and 6 were chosen because the normal stresses were the largestin those bays. Figure 10 shows that the chordwise distribution ofthe stringer stresses in the cover is approximately linear; thereby’tie use of.the theoretical equivalent-area coefficient of one-sixthappears to be justified.,. 1

Comparison of test results with theoretical cugy~t- A comparisonof the observed and calculated shear flow and normal flange stressesfor,both loading conditions c% be seen in figures 11 to 14. Thestre&ses were calculated as shown in the appendix %y means of the .“formulas presented in this.paper.

“ Examination of figures 11 &d 12 shows very good agreementbetween the observed end oaldhlatbd shear-stiess values, The onlypoint iq the tip loading oondition (fig. 11) that does not fallon theoalculated curve within the acc-cy of the Tuckerman @gereadings is a point on the spar 7.5 inches from the tip, which isapproximately 5 percent greater than the corresl,ondingcalculatedvalu9. This deviation,from the calculated curve is probably oaused .by the fact that the.section at which this measurement,is taken is,near the tip of the box.where the 10%I ig being applied. ‘.

IIkaminqti.onof figures,13 and 14 shows Bood a~~eement betweenthe”expe~imental =d oalctia~pd values of ‘he axial loads in tieflange due to torsion- For both loading conditions, the observed

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NACA ~ No ● 3.297 23

values in bays 5 and 6, whi~h include the most critical loads (theloads at the root of the box), agree within the accuracy of theTuckerman gage reading. The observed values in bays-3 and 4 comparefavorably @.th the calculated values for the tip-loading condition.For the quarter‘point loading condition, however, the olservedvalues for the test points in the victiity of bulkhead 3 areslfghtly greater than the calculated values for the same yoints.This deviation from the calculated values can be explained by theuncertainty of the conditions at the yotit of loading. The extentto which the loading fixture restrains the flanges &com warpingand the fact that the torque applied to the box through the sparwebs needs an appreciable distance to be distributed are only theobvious reasons for deviations to occur in the vicinity of theloading bulkheads. Also for these reasons, no attempt was madeto evaluate the stresses in the tip bay for me tip loading condition.

The values of the experimental stresses in figures 13 and 14,taken within 5 inches of bulkheads 1 and 2, are somewhat greaterthan the calculated values. This deviation may be explained inpart by local bending of the flange. The flange acts as acontinuous beam supported by an elastic foundation and loaded ateach bulkhead. The bending moment in this %eem is tie greatestat the buUsheads where the deviatim of the o%served values from thecalculated values are the greatest.

CONCLUDING REMARKS

The agreement that was obtained between the experimentallending stresses due to torsion in a @pered box %eam and theca.lculateavalues indicates that the metiod presented can be usedto obtain the mending-stress distribution in a tapered box beamunder torsion.

For boxes with very small taper the flange stresses in theoutboard half of the box we very small. In these boxes a firstapproximation of the bending stresses due to torsion can be madeby using $he properties of the tapered box at tie root, byconsidering the box nontapered, and then using the method of

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24 MICA TN Noa X297

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ma~sis. descr?vqd by ?aul @hn in.his ,paperon bending stressesdue tp. toys$pn .,.{~CA ARR,,WC, 2942). For mere ‘accuratecalculations.ot.thebe~ld:lng,stresses due~’totoriion in a tapered tox, however, ,themetho~ presented he&in shouldbe.used.

,.,”. .

. . .

,.. .

,.

.,

., ,.

,.,...

. .

.. .,“. . .

. .!.

,., ” .

●

I.,. . . ..-. I

:., .,’

. . ..,,... . .... . ,,. ,

!., ..:.,

. .,..

,.,

,.,..’

$., ;,:...

..... “’l”- ‘.‘.’,. ,,. ,.>

, ,,.,‘.,

- ...,..,.:.. ,., ,

,1 :...;’:. . . .

. .. ...

. . .,. ..

..... ., . ..”. .-.. . ... .... . ., ,_- .,. .-

‘... r”.,..?’.,”,- ,,, :’,,.,..- . . .. . . . . .. .... . . .

.,

. . . .

.’

“.

I

25 ,

.:, . ... , API’ENDIX “:’.,’,.“.”.... .. ..’”, ,.,..

SOLUTION OF STRESS DISTR12XJTIONm @2EF@l BOX BEA&i

In order to give an illustration of the method, a completesolution of the shear and no- stress dig tiibution in thetapered box usOd as a test specimen is given.

, I

.,.1

.. ..,

. . .

,.

The actual over-all dimensi~ of the specimen are given intQurQ 7.” The dimensions at the root and tip cross sections exe ‘ i

given in $igme 6, together with the dimensions for the”equivalentstructure. The :<fective flange area is obtained.from the !equation

...

(Al} 1

where ~ ie the area of a flange angle a& AS is the area 1I

of a Coker stringer. The effective flange ereas are assumed to 1%0 concentra.teilat the points of intersection of the center lines

1

of the cover sheets and spar webs. Ail properties of the equivalent..bo~~.t@y@r,linesrlyfrm tip to root. The %OX is divided intosix bays. The geometrical ~opcrties for the inboard =a outboard 1

end of each bay are lisfed”in table 2. -.I

“.

The values of E end G used in these calculations are, ‘respectively, 10,600 ksi and .4jO00 ksi. For the first loadingcondition, a torque of 100,000 ‘inch~ounds is applied to tie tipof the lox and for the second loading condition a torque of 1

100.,000inch-pounds is applied to the middle buMhead of the box.t1

The warpingof the eq~ttons

.constants “p, J; and f“,calc@ated by means

(22), (26), “%d (31), .~d the wa~ing due to -

.“

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26 NACA TN NO. ~29T

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t

torsion wnT, calculated by the use of equation (17)2 are talmlated

in the following table. The tabulated wnT values are for the Ioti

at tip of box.

Bay

1

2

3

4

5

6

. .

Warying constants

fn

1.610 X IQ-6

I

-60.829 x 10 1.564x 10%

I ● 574

1*714 “ I *794

1.785

1“857

1 ● 511

1.585

1.660

.gok 1.739

1.011 “ 1.817

‘T‘n I

6k3gx 10-6

5848

5274

4&)3

4410’

ko83

.

For abox witi’six bays, the recurrence fomnula (equation (37))and the.equations for the boundary conditions (equations (38) md. (39))give the following six equations. ,

,,

.( )- p~+fe xp+J@*=wJ9?- W1

‘)T

82X1 - (3?’2+ f3 X2 : J3X3 = W3T- W2

1J3XQ - (P3 + ‘4 )X3 + J4X4

= W4T - W3T > (A2)

~4x3 - (P4 + fcj)x4 + J5X5” = W5T - whT

J5X4 - (15 + f~ )X5 + #@6 = ~6T - W5T

j6X5 - 4X6 = W6T J

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NACA TN No ● 3.297

F&om substitution of the values from theformulas of equation (A2), the following

27

preceding table into theset of simultaneous equations

is obtained for the f&st- tip loading c&dition:

‘1 X2 X3 Xk

-3 ●121 0.574 -591.574 -3.166 0.684 ‘ -574

.684 -3*307 0 .79k -471.794 -3.453 0.904 -393

●904 -30602 1o11. “%71,o11 -I.857 -4083

The solution of these equations gives

‘1= 244 X4 = 437

‘2= 292

‘5= 965

X3 = 307 X6 = 2724

These values give the flange loads at each bull&ead. In order toobtain the distribution of the flmge loaw be~een the b~ea~~ .the formula

obtained

Forthe span

and

froma summation of equations (X2) end

calculation of the distribution of theof the box, tie formulas

(A3 )

(13) is used.

shear flow.along

Qcx (. ‘%’%”1 1 xn- %-1) ‘*~2 2L

!l~x= - ( )52kL&,~ -Xn.l +...z-2 2bcc.

> (A4)

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28 NACAm No ● 1297

‘,x.obtained from .asumnation of e@ations (J], (~), Ad (13) are used~:,A tabulation of:the fWnge loads “andshear flti~”can be seen,intable3. ... . .. .-’ ,f, . .!

“‘For the quarter’~oint loading condition the val~es’for themrQing constants p, ~, and f are the same as those for thetip’“lOadi~ condition● AMo the values for wnT’ are”.we same except

in bay~ ‘1;2, and 3 where wnT is zero. The forrnula~“(A2) can still,..be used for this loading condition. By substituting the a.ypropriatevalues In equation (A2), the following det of stiul.taneousequationsis obtq$ned for ,thequa&er -point loading condition:

xl %’: X3 X4 X5 X6 %+1 ‘-wn

-3.X21 0.574 “ ,1 .“ “ ::’ ‘“ “’ .’ o’--.574 -3 ●166 o.684

.684 -3.307 0 ● 79k 4&3● 79k .-3 s453 0.904 -393

*904 -3.602 1.011 -3Q7.. 1.011 -1.’857 -4083

The solution of these equations gives

‘1 z -63. . . . . . .Xti= -21 ,.: ‘,,.

.,, .. . .

%2’ -343 X5= ‘9 “ “

.“

= -1528 ,‘“3 ‘

X6 = 2650,,.’

.....

Again, by the use of the equations (A3) and (Ah) the distributionof the flange loads and shew flows was obtained along the span.The tabulations of these flsnge l-de and shear flows are ‘shownin txible4.

,... ,.:‘, ,.1 ,.. . .’ ::. ... .,

... .. . ..... .’

. ,’ . .1. . ,.. “ . .

I,

1

in

., ,.. ..,,

.

.

NRCA~ lb. 3297

BIMERfmm

1. Kuhn, Paul: A Method of Calculating Bendtng Stresses Due tuTorsion. NACA ARR, ~OC. 1942.

2. Ebner, Hans: Torsional StressesMartially Restrained against

.

in Bmr Wmts withW&rpzng*mm ~

Crms SectionsHo. 744, 1934.

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Seowfrl

RQOt

Tip:

tqe’rea

Moderatelytapered

Monla-d

TAME 1

ROOT AHDTIPDJMENSICPX30FM XERWS

(:. )

50.OcQ

20.000

35.m

50.(XXI

(A)

la .Ow

4.000

7Am

10.OfM

%

(In. )

0.064

.064

.064

.064

I I

(2*)o .07!2

●W

.072

●W

1.000 I 1.000

1.000 .200

1.000

I.200

1.000 ] .200

Iwl!IoIf4L ADvmaRYcokml!wEE m AERmmTIcs

I-. --

8

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.

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mu .l=W m,+

M3ZT “ON NJL Vi3VN

~ 0“

E$L

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~ACA ~ No. 1297 32

TABm 3

CAmm NORWLAm SEElm s~ m

TAPEREDM)XEE4M2KIRIQADAT~

[T = 100,000 in. -lb]

Distance q= *= q~ %from tip (?b) (;i) (lb/f.no) (psi) (xb/in.) (psi)

(in*)

o 0 0 334-9 47&l 317.1 5040

1 10 167 178 296.7 4240 2b.9 4470

15 244 257 -9”9 265.I “421O

15 244 257 273.8 3910 271.2 4310

2 25 269 277 245.0 3500 242.6 3&0

35 292 2gk 220.4 315Q 218.2 3470

35 292 294 219.7 3140 218.9 34??0

3 45 300 296 198.9 2840 198.1 3150

55 307 296 180.7 25@ la.1 2860

55 307 296 183.9 2630 176.9 28~

4“ 65 375 354 167.9 2400 161.5 2570

75 437 405 154.0 2200 148.o 2350

75 437 40~ 165.3 2360 136.7 2170

5 @ w 647 152.0 2170 125.8 2000

95’ 965 %9 U@.3 2000 U6.1 1~

95 965 “ %9 175.3 2500 81..1 Kgo

6 105 @8 1640 162.3 2320 75.1 llgo

11.5 2724 2336 150.8 2150 69.6 111o

NATIONALADVISORYCOhMITIEEFOR AERONATJT?>CS

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IlKACA TN NO. 1297 “i

TABIE 4

CALCULMEDMMMAL AND SBEARSTRESSESIIiTAPEREDBOX

BEAMEOR LOADAT QUARTERPfMNTOF SPAN

Distancew frlmltip -c~

(in.) (;b) (;i) (Ijim) (:i) (l~in.) (psi)

Io

“1 10

15

15

2 25

35

35

3 45

55

0.

-43

-63

-63

-211

-343

-343

-965

-1528

T=O

o

-46

-66

-66

-217

-345

:345

*1

-1473

=2.3

+!.0

-1.9

-7.8

-6:9

-6.3

-32.5

+g.k

-26.8

-30

-30

-30

2.3

,2.0

1.9

40

w30

-s!.0

-1oo

-90

-460

-420

-3@

7.8

6.9

6.3

9 ●5

2g.k

26.8

120

llo

100

520

430,

T . 100,000in.‘lb

55 -1528 -1473 221.5 3160 139.3 2210

4 ‘* -740 -@g 202.1 2890 X27.3 2020

75 *1 -19 185.4 2650 116.6 l&l

75 -21 -19 174.0 2450 “ 128.0 2030

5 % 421 “3& 160.0 22go u7.8 1870

95 829 738 147.8 2110 108.6 1730

95 829 738 1~.o 2530 79.4 1.260

6 105 lnk 1549 163.9 23ko 73”5 llm%

115 2650 2273 152.3 21a 68.1 1083

.

i

Ii

.

.

MMUONAL ADVISORYCOMMITTEEEURAERONATIKECS

7

I

< .

I

Bulkhead O

Figure 1.—

,,.-lI, __

——-

n-1

Convention for nwnberi~

1’1-.— --

txlys

,’1

and bulkheads,

Elz

w1

;1.’1, . _.

1’ — —

—

N

,,

.

---m—

t=

w

figure. 2 .—Free-bdy sketch oftypkdbay tl. %+,,

,“

t,-m

d .“,, . 1 .,, ~’”~:d:, ,,, “ .; - ‘~ —.— — ‘1 ,, ,L, “1 1 i,, ,,, >’”

A— —J. - ., .,,.,;”.,—— . -.— -.—- -, ——— .

. NACA TN No. 1297

.

.-.

.

.

. .

(a) Free-bed y sketch of section of spar web.-

%x~——

t

m“1 . .J t

c Clc 1 /qC+dqC C+ dc

I ti—-—l”

I-JClcx

dx

[O infinitesimal section of spar web-

.

.

Fig. 3

.NATIONAL ADVISORY

Co)lnl?-r.u ml US4NAVTICS

.

(c) Free-body sketch of flange.

figwc 3.-Free-bed] s~Ctch~ of ~~mwnent porf~ of spar.

~I

.,

t

i

.1

1

I

[

i

!

—- ;,

:1

.,

I

,

-,

, -.

..—

.i;

—-; i

;

i—..

,

II

—.-.

1.60

1,40

.

‘1

1.20

1.001.00 1.40

17gure 4,—Variation

-- -.—— — —— .— -

1.80 2.20

Taper ratio, R

of ta~er constants

‘..-—— —-. -

2.60

NATIONAL ADVISORY

COMMITTEE FOR AERONAUTICS

with taper ratio,

3.00

.40

‘.60

K2

]nd

K3

.80

00

-- -—- ——-—— -

. NACA TN NO. 1297

*

...

.

.

.

.

2400

2000

1600

.-lna.:g1200%al0c=k

800

400

0

A— High tape~ tapered flange area

B — High taper, cotwtont flange area “

C — Moderate taper, tapered flange Oreo

D— Moderate taper, constant flange area

E — No taper, constant flange area

Bulkhead 2

.,

4

Fig. 5a

/

Ill

Root

o 20 40 60 80 100 120

Distance from tip of box, ih.

(a) Flange stresses.

Figure 5—Comparison of stresses in boxes with

T= 100,000 inch-pounds.

NATIONU ADVISORYCOXNITTEE FM AERONAUTICS

varying tapen

I

i

-.

,.l

50

o

High taper, tapered flange area

— – — Moderate taper, tapered flange area

——— —tdo taper, tapered flange mea

o 20 40 60 80 100 120

MATK04AL MW$ORY

Distance from tip of box, in, cawllu m MGWJ’rfcs

(b) Sheet stresses.

Hgure 5.— Concluded,

!,

#-- ——

;.-.. —-—- ——, .—— ——— — ——. .

!3zp

. ..—

.’. .

Figure &- Setup for torsiontestsoftapered box Man

.—

m

, -. -

. .

. .. /.

~ @r?frOf

oxis

I ,.

~ ~ 3/4x3/4X3&? st?iw equal. Spacd

‘i9we ~ -~opekd

;,,--

-’ -,, ,

--’ ;

._

box b.ew. i

L

%g

+1.:, ,,,-- ..,! ‘.

:,, ,”,.,..; ,1 ;:,

J;,.1, ,

--_,

.— — .-

,.- m Root TiII

M .4CIUOI root md tip cross sections,

. .

A= 1.166 tb=0.0631:0.070

I :

ie

IN

●

(0o. —

+’- L0 m“ +

I)

●

I

L , 45.0704“ L

26,133

Root Tip

(b) Equiwdent mot ond tip cross sections. “ ~~~m~-~

F@HP. 8. —Actuol ord squivolent root md hp cross sectlona

# . . . . . ., ..,.“

I

Bay I Bay 4

Bay 2 Boy 5

Boy 3 Bay 6

Figure 9—OlslriWmn of shsor ‘SWSSS otij u&s seclbn. p la0,000 Inch-pamds,:

I III1’

1, :, ,“: ..,.—- .,— —

1

. .,’

. .-

\

0

,,

Flange

Stringer

2

3

4

5

6

‘LLG 7

c, ,s0

qe

/

G

c’

70

0

-\

o

)

Flange

Stringer

2

3

4

5

6

!/7

80

0 Flonqe

Section B

/’)

Section

AB

EBEaStringer

01234

, 1 I

Sale in 1000 psi

,

NmC+ML mvmlrfcmrrnEEm~D

Fiwe 10.— Dkltiktkm of strinqer stresses over mm section. T= IOO,OOO irmh-pwn~.

*:,

I I-.-——. —— — - —. . . . . . . ——.

.

‘1

1

I

,

.

NACA TN NO. 1297 Fig. 11

I

—-— Calculated Bredt shear flow

— Calculated spar shem flow

——— Galmlatecl cover shear ftow ‘-”

n Experim.cntal spar shear flow

o Experimental ccver shear flow

v1 ‘\I

o 20 40 60 80 too

Distance from tip of box, in.NATIO14AL ADVISORY

WHWTTSS FO! A!ROMAUTICS

I

-1

120

/

I1

1

Figure I l.— Comparison of cxperimcnfa! and calculated sheer flows for tlp Ioockg condlhon.i

T= [00,000 inch-pounds.I

I

Fig. 12 NACA TN ND. 1297

,I

I

240

200

160

e

s~- {20G=

hal$

80

40

0

-40

—— Calculated Bredt sheer flow

- Ga[culafed spor shear flow

——— Calculated cover shear flow

❑ Experimental spar shear flow

o Experimental cover shear flaw

.

--?P-1t

r——Y’

‘\\‘b,db,‘%7 ‘\

*

,

4

,

—

.-J

0 20 40 60 80 m 120.I

Distance from tip of box, in. NAT IONAL ADVISORYCONNITUE f~ 41RONAUTICS

Figure 12.— Comparison of experimental

loading condition.

~

,

●

ond calculated sheer flows for quarter-point

T= [00,000 inch-pounds. iI

1

●NACA TN No. 1297

,Fig. 13

240(

200(

8(3C

40(.

0

3ulkhead

— Calculated values

o Test points

/

01 0

0°

3 4 5-“”/.

o 20 40 60 80 100

NATIONAL ADVISORY

Distance from tip of box, in coMMtTIEEFolAERONAU7KS

-.

RI

—

It

-1

120

Figure 13.— Comparison of experimental and ~lcuhted flange loads fw tip loading condition.

I

1

1

I

I

,

1I

,,

I

I

I

1

I

.-

1

I1

II

I

I

I

I

I

T= 100,000 inch-pounds.

,

1

Fig. 14 NACA TN No. 1297

f

.

2800

2400

2000

1600

1200

~ 800

g-~

g ~wE

o

-400

-8cm

~

-12CK)

-1600

0

— Calculated values

o ‘Test points

1

01

0

0

4

. .

c

5

I

+

I“1

-1—

o 20 40 60 80 100 120

—-.-l-i

1ILKrlamu ADvlsOaY

~

Dktame fram lip of box, In, m#w-rsI FMUMIAVI!L54:

1II

Figure 14. — Gomporison cd cxpcrimentol and calculated flan9e Iaods fOr quorter- pointI

,

Ioadmg condttlon,

7

T= 100,000 inch-pounds. :